MOP Homework 2005-06 (MOSP) 39p

geometry problems from Math Olympiad Summer Program Homework (MOP)

with aops links in the names

2005 - 2006

2005 MOP Handout

2005 MOP Homework Red Group Geometry p1

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Point $D$ lies on arc $\frown{BC}$ not containing $A$. Let $E$ be the foot of perpendicular from $A$ to line $CD$. Prove that $BC+DC=2DE$.

2005 MOP Homework Red Group Geometry p2
Let $ABC$ be a triangle, and let $D$ be a point on side $AB$. Circle $\omega_1$ passes through $A$ and $D$ and is tangent to line $AC$ at $A$. Circle $\omega_2$ passes through $B$ and $D$ and is tangent to line $BC$ at $B$. Circles $\omega_1$ and $\omega_2$ meet at $D$ and $E$. Point $F$ is the reflection of $C$ across the perpendicular bisector of $AB$. Prove that points $D$, $E$, and $F$ are collinear.

2005 MOP Homework Red Group Geometry p3
Let $M$ be the midpoint of side $BC$ of triangle $ABC$ ($AB>AC$), and let $AL$ be the bisector of the angle $A$. The line passing through $M$ perpendicular to $AL$ intersects the side $AB$ at the point $D$. Prove that $AD+MC$ is equal to half the perimeter of triangle $ABC$.

2005 MOP Homework Red Group Geometry p4
Let $ABC$ be an obtuse triangle with $\angle A>90^{\circ}$, and let $r$ and $R$ denote its inradius and circumradius. Prove that $$\frac{r}{R} \le \frac{a\sin A}{a+b+c}.$$

2005 MOP Homework Red Group Geometry p5
Let $ABC$ be a triangle. Points $D$ and $E$ lie on sides $BC$ and $CA$, respectively, such that $BD=AE$. Segments $AD$ and $BE$ meet at $P$. The bisector of angle $BCA$ meet segments $AD$ and $BE$ at $Q$ and $R$, respectively. Prove that $\frac{PQ}{AD}=\frac{PR}{BE}$.

2005 MOP Homework Red Group Geometry p6
Consider the three disjoint arcs of a circle determined by three points of the circle. We construct a circle around each of the midpoint of every arc which goes the end points of the arc. Prove that the three circles pass through a common point.

2005 MOP Homework Red Group Geometry p7
Let $ABC$ be a triangle. Prove that $$\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} \ge 4\left(\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}\right).$$


2005 MOP Homework Blue Group Geometry p1

In convex hexagon $ABCDEF$ all sides have equal length and $\angle A+\angle C+\angle E=\angle B+\angle D+\angle F$. Prove that the diagonals $AD,BE,CF$ are concurrent.

2005 MOP Homework Blue Group Geometry p2
In a convex quadrilateral $ABCD$ the points $P$ and $Q$ are chosen on the sides $BC$ and $CD$ respectively so that $\angle{BAP}=\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ABP$ and $ADQ$, is perpendicular to $AC$ if and only if the triangles $ABP$ and $ADQ$ have the same areas.

2005 MOP Homework Blue Group Geometry p3
Circles $S_1$ and $S_2$ meet at points $A$ and $B$. A line through $A$ is parallel to the line through the centers of $S_1$ and $S_2$ and meets $S_1$ and $S_2$ again $C$ and $D$ respectively. Circle $S_3$ having $CD$ as its diameter meets $S_1$ and $S_2$ again at $P$ and $Q$ respectively. Prove that lines $CP$, $DQ$, and $AB$ are concurent.

2005 MOP Homework Blue Group Geometry p4
The incenter $O$ of an isosceles triangle $ABC$ with $AB=AC$ meets $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. Let $N$ be the intersection of lines $OL$ and $KM$ and let $Q$ be the intersection of lines $BN$ and $CA$. Let $P$ be the foot of the perpendicular from $A$ to $BQ$. If we assume that $BP=AP+2PQ$, what are the possible values of $\frac{AB}{BC}$?

2005 MOP Homework Blue Group Geometry p5 (+ Black 5)
Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$. Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$.

2005 MOP Homework Blue Group Geometry p6
A circle which is tangent to sides $AB$ and $BC$ of triangle $ABC$ is also tangent to its circumcircle at point $T$. If $I$ in the incenter of triangle $ABC$, show that $\angle ATI=\angle CTI$.

2005 MOP Homework Blue Group Geometry p7
Points $E$, $F$, $G$, and $H$ lie on sides $AB$, $BC$, $CD$, and $DA$ of a convex quadrilateral $ABCD$ such that $\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA}=1$. Points $A$, $B$, $C$, and $D$ lie on sides $H_1E_1$, $E_1F_1$, $F_1G_1$, and $G_1H_1$ of a convex quadrilateral $E_1F_1G_1H_1$ such that $E_1F_1 \parallel EF$, $F_1G_1 \parallel FG$, $G_1H_1 \parallel GH$, and $H_1E_1 \parallel HE$. Given that $\frac{E_1A}{AH_1}=a$, express $\frac{F_1C}{CG_1}$ in terms of $a$.


2005 MOP Homework Black Group Geometry p1
A circle with center $O$ is tangent to the sides of the angle with the vertex $A$ at the points B and C. Let M be a point on the larger of the two arcs $BC$ of this circle (different from $B$ and $C$) such that $M$ does not lie on the line $AO$. Lines $BM$ and $CM$ intersect the line $AO$ at the points $P$ and $Q$ respectively. Let $K$ be the foot of the perpendicular drawn from $P$ to $AC$ and $L$ be the foot of the perpendicular drawn from $Q$ to $AB$. Prove that the lines $OM$ and $KL$ are perpendicular.

2005 MOP Homework Black Group Geometry p2
Let $I$ be the incenter of triangle $ABC$, and let $A_1$, $B_1$, and $C_1$ be arbitrary points lying on segments $AI$,$BI$, and $CI$, respectively. The perpendicular bisectors of segments $AA_1$, $BB_1$, and $CC_1$ form triangles $A_2B_2C_2$. Prove that the circumcenter of triangle $A_2B_2C_2$ coincides with the circumcenter of triangle $ABC$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$.

2005 MOP Homework Black Group Geometry p3
Points $M$ and $M'$ are isogonal conjugates in the traingle $ABC$. We draw perpendiculars $MP$, $MQ$, $MR$, and $M'P'$, $M'Q'$, $M'R'$ to the sides $BC$, $AC$, $AB$ respectively. Let $QR$, $Q'R'$, and $RP$, $R'P'$ and $PQ$, $P'Q'$ intersect at $E$, $F$, $G$ respectively. Show that the lines $EA$, $FB$, and $GC$ are parallel.

2005 MOP Homework Black Group Geometry p4
Let $ABCD$ be a convex quadrilateral and let $K$, $L$, $M$, $N$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Let $NL$ and $KM$ meet at point $T$. Show that $8[DNTM] < [ABCD] < 8[DNTM]$, where $[P]$ denotes area of $P$.

2005 MOP Homework Black Group Geometry p5 (+ Blue 5)
Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$. Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$.

2005 MOP Homework Black Group Geometry p6
Given a convex quadrilateral $ABCD$. The points $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$ respectively. The line $PQ$ intersects the lines $AB$ and $CD$ at $N$ and $M$ respectively. Prove that the circumcircles of triangles $NAP$, $NBQ$, $MQD$, and $MPC$ have a common point.

2005 MOP Homework Black Group Geometry p7
Let $ABCD$ be a cyclic quadrilateral who interior angle at $B$ is $60$ degrees. Show that if $BC=CD$, then $CD+DA=AB$. Does the converse hold?

2006 MOP Handout

2006 MOP Homework Red Group Geometry p1

$ABC$ is an acute triangle. The points $B'$ and $C'$ are the reflections of $B$ and $C$ across the lines $AC$ and $AB$ respectively. Suppose that the circumcircles of triangles$ABB$' and $ACC'$ meet at $A$ and $P$. Prove that the line $PA$ passes through the circumcenter of triangle$ABC.$

2006 MOP Homework Red Group Geometry p2

In triangle $ABC$,$\angle BAC = 120^o$. Let the angle bisectors of angles $A;B$and $C$ meet the opposite sides at $D;E$ and$F$ respectively. Prove that the circle on diameter $EF$ passes through $D.$

2006 MOP Homework Red Group Geometry p3

Let P and Q be interior points of triangle ABC such that \ACP = \BCQ and \CAP = \BAQ. Denote by D,E and F the feet of the perpendiculars from P to the lines BC, CA and AB, respectively. Prove that if \DEF = 90, then Q is the orthocenter of triangle BDF.

2006 MOP Homework Red Group Geometry p4

Let $ABC$ be a right triangle with$\angle A = 90^o$. Point $D$ lies on side $BC$ such that $\angle BAD = \angle CAD$. Point $I_a$ is the excenter of the triangle opposite $A$. Prove that $\frac{AD}{DI_a } \le \sqrt{2} -1$

2006 MOP Homework Red Group Geometry p5

Show that among the vertices of any area $1$ convex polygon with $n > 3$ sides there exist four such that the quadrilateral formed by these four has area at least $1/2$.

2006 MOP Homework Blue Group Geometry p1

Triangle $ABC$ is inscribed in circle $w$. Line $l_{1}$ bisects $\angle BAC$ and meets segments $BC$ and $w$ in $D$ and $M$,respectively. Let $y$ denote the circle centered at $M$ with radius $BM$. Line $l_{2}$ passes through $D$ and meets circle $y$ at $X$ and $Y$. Prove that line $l_{1}$ also bisects $\angle XAY$

2006 MOP Homework Blue Group Geometry p2

Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.

2006 MOP Homework Blue Group Geometry p3

Let $ABC$ be a triangle with $AB\neq AC$, and let $A_{1}B_{1}C_{1}$ be the image of triangle $ABC$ through a rotation $R$ centered at $C$. Let $M,E , F$ be the midpoints of the segments $BA_{1}, AC, BC_{1}$ respectively. Given that $EM = FM$, compute $\angle EMF$.

2006 MOP Homework Blue Group Geometry p4

Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively. Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent if and only if $AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$

2006 MOP Homework Blue Group Geometry p5

Let $ABCD$ be a convex quadrilateral. Lines $AB$ and $CD$ meet at $P$, and lines $AD$ and $BC$ meet at $Q$. Let $O$ be a point in the interior of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that $\angle AOB  +\angle  COD = 180$.

2006 MOP Homework Blue Group Geometry p6

In triangle $ABC, AB \ne AC$. Circle $\omega$ passes through $A$ and meets sides $AB$ and $AC$ at $M$ and $N$, respectively, and the side $BC$ at $P$ and $Q$ such that $Q$ lies in between $B$ and $P$. Suppose that $MP //  AC, NQ // AB$, and $BP \cdot  AC = CQ \cdot   AB$. Find $\angle BAC$.

2006 MOP Homework Blue Group Geometry p7

In acute triangle $ABC, CA \ne BC$. Let $I$ denote the incenter of triangle $ABC$. Points $A_1$ and $B_1$ lie on rays $CB$ and $CA$, respectively, such that $2CA_1 = 2CB_1 = AB + BC + CA$. Line $CI$ intersects the circumcircle of triangle $ABC$ again at $P$ (other than $C$). Point $Q$ lies on line $AB$ such that $PQ \perp CP$. Prove that $QI \perp A_1B_1$.

2006 MOP Homework Black Group Geometry p1

In isosceles triangle $ABC$, $AB=AC$. Extend segment $BC$ through $C$ to $P$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, such that $PX \parallel AC$ and $PY \parallel AB$. Point $T$ lies on the circumcircle of triangle $ABC$ such that $PT \perp XY$. Prove that $\angle BAT = \angle CAT$.

2006 MOP Homework Black Group Geometry p2

Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.

2006 MOP Homework Black Group Geometry p3

There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps.

2006 MOP Homework Black Group Geometry p4

Let $ABC$ be a triangle with circumcenter $O$. Let $A_1$ be the midpoint of side $BC$. Ray $AA_1$ meet the circumcircle of triangle $ABC$ again at $A_2$ (other than A). Let $Q_a$ be the foot of the perpendicular from $A_1$ to line $AO$. Point $P_a$ lies on line $Q_aA_1$ such that $P_aA_2 \perp A_2O$. Define points $P_b$ and $P_c$ analogously. Prove that points $P_a, P_b$, and $P_c$ lie on a line.

2006 MOP Homework Black Group Geometry p5

Let $ABC$ be an acute triangle with $AC \neq BC$. Points $H$ and $I$ are the orthocenter and incenter of the triangle, respectively. Line $CH$ and $CI$ meet the circumcircle of triangle $ABC$ again at $D$ and $L$ (other than $C$), respectively. Prove that $\angle CIH=90^{\circ}$ if and only if $\angle IDL=90^{\circ}$.

2006 MOP Homework Black Group Geometry p6

Let $P$ be a convex polygon in the plane. A real number is assigned to each point in the plane so that the sum of the numbers assigned to the vertices of any polygon similar to $P$ is equal to $0$. Prove that all the assigned numbers are equal to $0$.

2006 MOP Homework Black Group Geometry p7

Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $T$. Let $X$ be a point on circle $\omega_1$. Line $l_1$ is tangent to circle $\omega_1$ and $X$, and line $l$ intersects circle $\omega_2$ at $A$ and $B$. Line $XT$ meets circle $\omega$ at $S$. Point $C$ lies on arc $TS$ (of circle $\omega_2$, not containing points $A$ and $B$). Point $Y$ lies on circle $\omega_1$ and line $YC$ is tangent to circle $\omega_1$. Let $I$ be the intersection of lines $XY$ ad $SC$. Prove that...

a) points $C$, $T$, $Y$, $I$ lie on a circle

b) $I$ is an excenter of triangle $ABC$.

1997 - 1998 under construction

1997 MOP  (also, also, also)

Let $Q$ be a quadrilateral whose side lengths are $a, b, c, d$ in that order. Show that the area of $Q$ does not exceed $(ac+bd)/2$.

1997 MOP 

Let P be a point in the plane of a triangle ABC. A circle Γ passing through P intersects the circumcircles of triangles P BC, P CA, P AB at A1, B1, C1, respectively, and lines P A, P B, P C intersect Γ at A3, B3, C3. Prove that: the lines A1A2, B1B2, C1C2 are concurrent 

1997 MOP  (also, also)  [MOP97/2/5]

Let $ABC$ be a triangle, and let $D,E,F$ be points where the incircle touches sides $BC, CA, AB,$ respectively. The parallel to $AB$ through $E$ meets $DF$ at $Q$, and the paralleled to $AB$ through $D$ meets $EF$ at T. Prove that the lines $CF, DE, QT$ are concurrent.

1997 MOP 

Let $ABC$ be a triangle, and let $A',B',C'$ be the midpoints of the arcs $BC,CA,AB,$ respectively, of the circumcircle of $ABC$. The line $A'B'$ meets $BC$ and $AC$ at $S$ and $T$. $B'C'$ meets $AC$ and $AB$ at $F$ and $P$, and $C'A'$ meets $AB$ and $BC$ at $Q$ and $R$. Prove that the segments $PS,QT,FR$ concur.

1997 MOP (also, also)

Let $ABC$ be a triangle and let $O$ be its circumcenter. The lines $AB$ and $AC$ meet the circumcircle of triangle $BOC$ again at $B_1$ and $C_1$, respectively. Let $D$ be the intersection of lines $BC$ and  $B_1C_1$. Show that the circle tangent to $AD$ at $A$ and having its center on $B_1C_1$ is orthogonal to the circle with diameter $OD$.

1997 MOP (also, also, also)

Consider a triangle $AMC$ with $AB = AC$ and points $M, N$ lie on $AB$, $AC$, respectively. The lines $BN$ and $CM$ intersect at P. Prove that $MN$ and $BC$ are parallel iff $\angle{APM} = \angle{APN}$.

1997 MOP (?)

Let circle $\omega_1$, centered at $O_1$, and circle $\omega_2$, centered at $O_2$, meet at A and B. A line through A meets $\omega_1$ again at Y and meets $\omega_2$ again at Z. Let X be the intersection of the tangent to $\omega_1$ at Y and the tangent to $\omega_2$ at Z. Let $\omega$ be the circumcircle of $O_1O_2B$, and let Q be the second intersection of $\omega$ with BX. Prove that the length of XQ equals the diameter of $\omega$.

1998 MOP  [MOP98/4/5] (IMO Shortlist 1997, Q9)

Let $A_{1}A_{2}A_{3}$ be a non-isosceles triangle with the incenter $I$. Let $C_{i}, i = 1, 2, 3$, be the smaller circle through $I$ tangent to both $A_{i}A_{i+1}$ and $A_{i}A_{i+2}$ (summation of indices is done modulus 3). Let $B_{i}, i = 1, 2, 3$, be the other intersection point of $C_{i+1}$ and $C_{i+2}$. Prove that the circumcenters of the triangle $A_{1}B_{1}I$, $A_{2}B_{2}I$, $A_{3}B_{3}I$ are collinear.

1998 MOP  [MOP98/12/3]
Let ω1 and ω2 be two circles of the same radius, intersecting at A and B. Let O be the midpoint of AB. Let CD be a chord of ω1 passing through O, and let the segment CD meet ω2 at P . Let EF be a chord of ω2 passing through O, and let the segment EF meet ω1 at Q. Prove that AB, CQ, EP are concurrent.

1998 MOP [MOP98/5/4]
The bisectors of angles A, B, C of triangle ABC meet its circumcircle again at the points K, L, M, respectively. Let R be an internal point on side AB. The points P and Q are defined by the conditions: RP is parallel to AK and BP is perpendicular to BL; RQ is parallel to BL and AQ is perpendicular to AK. Show that the lines KP, LQ, MR concur.

1998 MOP  (also, also)
Let $ABCDEF$ be a cyclic hexagon with $AB=CD=EF$. Prove that the intersections of $\overline{AC}$ with $\overline{BD}$, of $\overline{CE}$ with $\overline{DF}$, and of $\overline{EA}$ with $\overline{FB}$ form a triangle similar to $\triangle BDF$.

1998 MOP 
If ABC is an acute triangle with circumcenter O, orthocenter H and circumradius R, show that for any point P on the segment OH, $PA+PB+PC \le  3R$.